Induced subgraphs of graphs with large chromatic number. XIII. New brooms

Alex Scott, Paul Seymour

Research output: Contribution to journalArticle

Abstract

Gyárfás (1975) and Sumner (1981) independently conjectured that for every tree T, the class of graphs not containing T as an induced subgraph is χ-bounded, that is, the chromatic numbers of graphs in this class are bounded above by a function of their clique numbers. This remains open for general trees T, but has been proved for some particular trees. For k≥1, let us say a broom of length k is a tree obtained from a k-edge path with ends a,b by adding some number of leaves adjacent to b, and we call a its handle. A tree obtained from brooms of lengths k1,…,kn by identifying their handles is a (k1,…,kn)-multibroom. Kierstead and Penrice (1994) proved that every (1,…,1)-multibroom T satisfies the Gyárfás–Sumner conjecture, and Kierstead and Zhu (2004) proved the same for (2,…,2)-multibrooms. In this paper we give a common generalization; we prove that every (1,…,1,2,…,2)-multibroom satisfies the Gyárfás-Sumner conjecture.

Original languageEnglish (US)
Article number103024
JournalEuropean Journal of Combinatorics
Volume84
DOIs
StatePublished - Feb 2020

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Induced Subgraph
Chromatic number
Graph in graph theory
Clique number
Leaves
Adjacent
Path
Class

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics

Cite this

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title = "Induced subgraphs of graphs with large chromatic number. XIII. New brooms",
abstract = "Gy{\'a}rf{\'a}s (1975) and Sumner (1981) independently conjectured that for every tree T, the class of graphs not containing T as an induced subgraph is χ-bounded, that is, the chromatic numbers of graphs in this class are bounded above by a function of their clique numbers. This remains open for general trees T, but has been proved for some particular trees. For k≥1, let us say a broom of length k is a tree obtained from a k-edge path with ends a,b by adding some number of leaves adjacent to b, and we call a its handle. A tree obtained from brooms of lengths k1,…,kn by identifying their handles is a (k1,…,kn)-multibroom. Kierstead and Penrice (1994) proved that every (1,…,1)-multibroom T satisfies the Gy{\'a}rf{\'a}s–Sumner conjecture, and Kierstead and Zhu (2004) proved the same for (2,…,2)-multibrooms. In this paper we give a common generalization; we prove that every (1,…,1,2,…,2)-multibroom satisfies the Gy{\'a}rf{\'a}s-Sumner conjecture.",
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Induced subgraphs of graphs with large chromatic number. XIII. New brooms. / Scott, Alex; Seymour, Paul.

In: European Journal of Combinatorics, Vol. 84, 103024, 02.2020.

Research output: Contribution to journalArticle

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